In 16-18, use the techniques of Example 8.4.4 and Example 8.4.5 to find the given numbers.

675 307   mod   713

To calculate:

The value of 675307(mod713).

Given information:

The given expression is 675307(mod713).

Formula used:

Division algorithm:

Let a be an integer and d be a positive integer. Then, there are unique integers q and r with 0≤r<d such that a=dq+r, here, q is the quotient and r is the remainder.

So, q=a div d and r=amodd.

Know that the technique is xm(mody)=[x(mody)]m(mody) and abmodn=[(amodn)(bmodn)](modn).

Calculation:

Write the 307 as a sum of powers of 2.

307=256+32+16+2+1

First, find the values of 6751(mod713),6752(mod713),6754(mod713),6758(mod713),

67516(mod713),67532(mod713),67564(mod713),675128(mod713) and 675256(mod713).

To find the value of 6751(mod713), use the formula x2(mody)=[x(mody)]2(mody).

6751(mod713)=675(mod713)=675

Thus, the value of 6751(mod713) is 675.

To find the value of 6752(mod713), use the formula x2(mody)=[x(mody)]2(mody).

6752(mod713)=455625(mod713) [455625=639⋅713+18]=18

Thus, the value of 6752(mod713) is 18.

To find the value of 6754(mod713), use the formula x2(mody)=[x(mody)]2(mody).

6754(mod713)=( 675 2)2(mod713)=[ 6752( mod713)]2(mod713)=[455625( mod55)]2(mod713)=182(mod713)

Simplify the above equation.

182(mod713)=324(mod713)=324

Thus, the value of 6754(mod713) is 324.

To find the value of 6758(mod713), use the formula x2(mody)=[x(mody)]2(mody).

6758(mod713)=( 675 4)2(mod713)=[ 6754( mod713)]2(mod713)=(324)2(mod713)

Simplify the above equation.

(324)2(mod713)=104976(mod713) [104976=147⋅713+165]=165

Thus, the value of 6758(mod713) is 165.

To find the value of 67516(mod713), use the formula x2(mody)=[x(mody)]2(mody).

67516(mod713)=( 675 8)2(mod713)=[ 6758( mod713)]2(mod713)=(165)2(mod713)

Simplify the above equation.

(165)2(mod713)=27225(mod713) [27225=38⋅713+131]=131

Thus, the value of 67516(mod713) is 131.

To find the value of 67532(mod713), use the formula x2(mody)=[x(mody)]2(mody)